The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

N.B. Due to a problem with the microphone the audio for the first 4 minutes of this recording is missing.

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

In the last 30 years, random combinatorial structures and their limits have been a flourishing area of research at the interface between probability and combinatorics. In this minicourse, I hope to show you some of the beautiful theory that arises when considering scaling limits of random trees and graphs.

Trees are fundamental objects in combinatorics and the enumeration of different classes of trees is a classical subject. In the first part of the minicourse, we will take as our basic object the genealogical tree of a critical Galton-Watson branching process. (As well as having nice probabilistic properties, this class turns out to include various natural types of random combinatorial tree in disguise.) In the same way as Brownian motion is the universal scaling limit for centred random walks of finite step-size variance, it turns out that all critical Galton-Watson trees with finite offspring variance have a universal scaling limit, Aldous' Brownian continuum random tree. In the infinite variance case, assuming certain tail conditions for the offspring distribution, other scaling limits arise, the so-called stable trees.

The simplest model of a random network is the Erdős-Rényi random graph: we take n vertices, and include each possible edge independently with probability p. One of the most well-known features of this model is that it undergoes a phase transition. Take p=c/n. Then for c<, the components have size O(log n), whereas for c>1, there is a giant component, comprising a positive fraction of the vertices, and a collection of O(log n) components. In the second part of this minicourse, we will focus on the critical setting, c=1, where the largest components have size on the order n2/3, and are "close" to being trees, in the sense that they have only finitely many more edges. We will see how to use a comparison with a branching process in order to derive the scaling limit of the critical Erdős-Rényi random graph.

Time permitting, we will then move on to consider the more general setting of a critical random graph generated according to the configuration model with independent and identically distributed degrees. Here, it is possible to obtain the same scaling limit as in the Erdős-Rényi case, but also others related to the stable trees.

The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases).
It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist.
The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

Over there do not exist any non-zero abelian varieties with good reduction everywhere. However, over all but six real quadratic fields they actually do exist. In this talk we determine, for certain small real quadratic fields , the set of abelian varieties over with good reduction everywhere.

Networks describe how things are connected, and are ubiquitous in science and society. Networks can be very concrete, like road networks connecting cities or networks of wires connecting computers. They can represent more abstract connections such as friendship on Facebook. Networks are widely used to model connections between things that have no real connections. For example, Biologists try to understand how cells work by studying networks connecting proteins that interact with each other, and Economists try to understand markets by studying networks connecting institutions that trade with each other.
Questions we ask about a network include "which components of the network are the most important?", "how well do things like information, cars, or disease spread through the network?", and "does the network have a governing structure?".
Professor Spielman will explain how mathematicians address these questions by modeling networks as physical objects, imagining that the connections are springs, electrical resistors, or pipes that carry fluid, and analyzing the resulting systems.

Networks describe how things are connected, and are ubiquitous in science and society. Networks can be very concrete, like road networks connecting cities or networks of wires connecting computers. They can represent more abstract connections such as friendship on Facebook. Networks are widely used to model connections between things that have no real connections. For example, Biologists try to understand how cells work by studying networks connecting proteins that interact with each other, and Economists try to understand markets by studying networks connecting institutions that trade with each other.
Questions we ask about a network include "which components of the network are the most important?", "how well do things like information, cars, or disease spread through the network?", and "does the network have a governing structure?".
Professor Spielman will explain how mathematicians address these questions by modeling networks as physical objects, imagining that the connections are springs, electrical resistors, or pipes that carry fluid, and analyzing the resulting systems.